(* Content-type: application/vnd.wolfram.mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 8.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       157,          7]
NotebookDataLength[      9528,        334]
NotebookOptionsPosition[      8481,        292]
NotebookOutlinePosition[      8819,        307]
CellTagsIndexPosition[      8776,        304]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{

Cell[CellGroupData[{
Cell[BoxData[
 RowBox[{"Solve", "[", 
  RowBox[{
   RowBox[{"2", "*", "Pi", "*", 
    RowBox[{"Integrate", "[", 
     RowBox[{
      RowBox[{
       RowBox[{"n", "/", 
        RowBox[{"(", 
         RowBox[{"4", "*", "Pi"}], ")"}]}], "*", 
       RowBox[{"Sin", "[", "\[Theta]", "]"}]}], ",", 
      RowBox[{"{", 
       RowBox[{"\[Theta]", ",", "\[Theta]0", ",", "Pi"}], "}"}]}], "]"}]}], 
   "\[Equal]", 
   RowBox[{"n", "-", "1"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.515498799627646*^9, 3.515498840148065*^9}, {
  3.515498871607711*^9, 3.515498889029787*^9}, {3.515773956498927*^9, 
  3.515773957474042*^9}}],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"{", 
   RowBox[{
    RowBox[{"Cos", "[", "\[Theta]0", "]"}], "\[Rule]", 
    FractionBox[
     RowBox[{
      RowBox[{"-", "2"}], "+", "n"}], "n"]}], "}"}], "}"}]], "Output",
 CellChangeTimes->{
  3.515498840851452*^9, {3.515498884441114*^9, 3.515498889547843*^9}, 
   3.515773959138673*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\[Theta]0", " ", "=", " ", 
  RowBox[{"ArcCos", "[", 
   RowBox[{
    RowBox[{"(", 
     RowBox[{"n", "-", "2"}], ")"}], "/", "n"}], "]"}]}]], "Input",
 CellChangeTimes->{{3.515498896913373*^9, 3.515498910706205*^9}}],

Cell[BoxData[
 RowBox[{"ArcCos", "[", 
  FractionBox[
   RowBox[{
    RowBox[{"-", "2"}], "+", "n"}], "n"], "]"}]], "Output",
 CellChangeTimes->{3.515498911709049*^9, 3.515773961950819*^9}]
}, Open  ]],

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"r", "[", "\[Theta]_", "]"}], " ", ":=", " ", 
   RowBox[{"Sqrt", "[", 
    RowBox[{"2", "-", 
     RowBox[{"2", " ", 
      RowBox[{"Cos", "[", "\[Theta]", "]"}]}]}], "]"}]}], ";"}]], "Input",
 CellChangeTimes->{{3.515505902514844*^9, 3.515505918500266*^9}, {
  3.515506054758475*^9, 3.515506063285266*^9}}],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"H1", " ", "=", " ", 
  RowBox[{"FullSimplify", "[", 
   RowBox[{
    RowBox[{"2", "*", "Pi", "*", 
     RowBox[{"Integrate", "[", 
      RowBox[{
       RowBox[{
        RowBox[{"n", "/", 
         RowBox[{"(", 
          RowBox[{"4", " ", "Pi"}], ")"}]}], " ", 
        RowBox[{"1", "/", 
         RowBox[{"r", "[", "\[Theta]", "]"}]}], " ", 
        RowBox[{"Sin", "[", "\[Theta]", "]"}]}], ",", 
       RowBox[{"{", 
        RowBox[{"\[Theta]", ",", "\[Theta]0", ",", "Pi"}], "}"}]}], "]"}]}], 
    " ", ",", 
    RowBox[{"n", ">", "0"}]}], "]"}]}]], "Input",
 CellChangeTimes->{{3.515505879960637*^9, 3.51550592343085*^9}, {
  3.515506022935377*^9, 3.515506112593429*^9}, {3.515506161165503*^9, 
  3.515506190452168*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"-", 
   SqrtBox["n"]}], "+", "n"}]], "Output",
 CellChangeTimes->{
  3.515506043665024*^9, {3.515506089654374*^9, 3.515506103328549*^9}, 
   3.515506151038624*^9, 3.515506182247937*^9, 3.515773988834234*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"H2", " ", "=", " ", 
  RowBox[{"FullSimplify", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"-", "2"}], "*", "Pi", "*", 
     RowBox[{"Integrate", "[", 
      RowBox[{
       RowBox[{
        RowBox[{"n", "/", 
         RowBox[{"(", 
          RowBox[{"4", " ", "Pi"}], ")"}]}], " ", 
        RowBox[{"Log", "[", 
         RowBox[{"r", "[", "\[Theta]", "]"}], "]"}], " ", 
        RowBox[{"Sin", "[", "\[Theta]", "]"}]}], ",", 
       RowBox[{"{", 
        RowBox[{"\[Theta]", ",", "\[Theta]0", ",", "Pi"}], "}"}]}], "]"}]}], 
    ",", 
    RowBox[{"n", ">", "0"}]}], "]"}]}]], "Input",
 CellChangeTimes->{{3.515506192936988*^9, 3.515506220120292*^9}, 
   3.515773968906846*^9}],

Cell[BoxData[
 RowBox[{
  FractionBox["1", "2"], " ", 
  RowBox[{"(", 
   RowBox[{
    RowBox[{"-", "1"}], "+", "n", "+", 
    RowBox[{"Log", "[", "4", "]"}], "-", 
    RowBox[{"n", " ", 
     RowBox[{"Log", "[", "4", "]"}]}], "-", 
    RowBox[{"Log", "[", "n", "]"}]}], ")"}]}]], "Output",
 CellChangeTimes->{3.515506233189875*^9, 3.515774003288511*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"H3", " ", "=", " ", 
  RowBox[{"FullSimplify", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"-", "2"}], "*", "Pi", "*", 
     RowBox[{"Integrate", "[", 
      RowBox[{
       RowBox[{
        RowBox[{"n", "/", 
         RowBox[{"(", 
          RowBox[{"4", " ", "Pi"}], ")"}]}], " ", 
        RowBox[{"Log", "[", 
         RowBox[{"2", "+", 
          RowBox[{"r", "[", "\[Theta]", "]"}]}], "]"}], " ", 
        RowBox[{"Sin", "[", "\[Theta]", "]"}]}], ",", 
       RowBox[{"{", 
        RowBox[{"\[Theta]", ",", "\[Theta]0", ",", "Pi"}], "}"}]}], "]"}]}], 
    ",", 
    RowBox[{"n", ">", "0"}]}], "]"}]}]], "Input",
 CellChangeTimes->{{3.515506252687989*^9, 3.515506284863824*^9}, 
   3.515773994818472*^9}],

Cell[BoxData[
 RowBox[{
  FractionBox["1", "2"], " ", 
  RowBox[{"(", 
   RowBox[{
    RowBox[{"-", "1"}], "+", 
    RowBox[{"2", " ", 
     SqrtBox["n"]}], "+", 
    RowBox[{"Log", "[", "4", "]"}], "-", 
    RowBox[{"n", " ", 
     RowBox[{"(", 
      RowBox[{"1", "+", 
       RowBox[{"Log", "[", "4", "]"}]}], ")"}]}], "-", 
    RowBox[{"2", " ", 
     RowBox[{"(", 
      RowBox[{
       RowBox[{"-", "1"}], "+", "n"}], ")"}], " ", 
     RowBox[{"Log", "[", 
      RowBox[{"1", "+", 
       FractionBox["1", 
        SqrtBox["n"]]}], "]"}]}]}], ")"}]}]], "Output",
 CellChangeTimes->{3.515506323892109*^9, 3.51577404282967*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"e", " ", "=", 
  RowBox[{
   RowBox[{
    RowBox[{"n", "/", "2"}], " ", "*", " ", 
    RowBox[{"(", 
     RowBox[{"H1", "+", "H2", "+", "H3"}], ")"}]}], "//", 
   "FullSimplify"}]}]], "Input",
 CellChangeTimes->{{3.515506328196653*^9, 3.515506336663474*^9}, {
  3.515506373281404*^9, 3.515506378554288*^9}, {3.515774165286677*^9, 
  3.515774184852973*^9}}],

Cell[BoxData[
 RowBox[{
  FractionBox["1", "4"], " ", 
  RowBox[{"(", 
   RowBox[{
    RowBox[{
     RowBox[{"-", "2"}], " ", 
     RowBox[{"(", 
      RowBox[{
       RowBox[{"-", "1"}], "+", "n"}], ")"}], " ", "n", " ", 
     RowBox[{"(", 
      RowBox[{
       RowBox[{"-", "1"}], "+", 
       RowBox[{"Log", "[", "4", "]"}], "+", 
       RowBox[{"Log", "[", 
        RowBox[{"1", "+", 
         FractionBox["1", 
          SqrtBox["n"]]}], "]"}]}], ")"}]}], "-", 
    RowBox[{"n", " ", 
     RowBox[{"Log", "[", "n", "]"}]}]}], ")"}]}]], "Output",
 CellChangeTimes->{{3.515774180179234*^9, 3.515774185240164*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"FullSimplify", "[", 
  RowBox[{
   RowBox[{"Series", "[", 
    RowBox[{"e", ",", 
     RowBox[{"{", 
      RowBox[{"n", ",", "Infinity", ",", "0"}], "}"}]}], "]"}], ",", 
   RowBox[{"n", ">", "0"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.515506470810012*^9, 3.515506511335747*^9}}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{
   RowBox[{
    RowBox[{"(", 
     RowBox[{
      FractionBox["1", "2"], "+", 
      RowBox[{"Log", "[", "2", "]"}]}], ")"}], " ", 
    SuperscriptBox["n", "2"]}], "-", 
   FractionBox[
    SuperscriptBox["n", 
     RowBox[{"3", "/", "2"}]], "2"], "+", 
   RowBox[{
    FractionBox["1", "4"], " ", 
    RowBox[{"(", 
     RowBox[{"1", "-", 
      RowBox[{"4", " ", 
       RowBox[{"Log", "[", "2", "]"}]}], "+", 
      RowBox[{"Log", "[", "n", "]"}]}], ")"}], " ", "n"}], "-", 
   FractionBox[
    SqrtBox["n"], "3"], "+", 
   FractionBox["1", "8"], "+", 
   InterpretationBox[
    SqrtBox[
     RowBox[{"O", "[", 
      FractionBox["1", "n"], "]"}]],
    SeriesData[$CellContext`n, 
     DirectedInfinity[1], {}, -4, 1, 2],
    Editable->False]}],
  SeriesData[$CellContext`n, 
   DirectedInfinity[1], {Rational[1, 2] + Log[2], 
    Rational[-1, 2], Rational[1, 4] (1 - 4 Log[2] + Log[$CellContext`n]), 
    Rational[-1, 3], 
    Rational[1, 8]}, -4, 1, 2],
  Editable->False]], "Output",
 CellChangeTimes->{{3.515506478077185*^9, 3.515506511801795*^9}}]
}, Open  ]],

Cell[BoxData[""], "Input",
 CellChangeTimes->{{3.515506579395879*^9, 3.515506580913939*^9}}]
},
WindowSize->{740, 607},
WindowMargins->{{203, Automatic}, {Automatic, 31}},
FrontEndVersion->"8.0 for Linux x86 (32-bit) (February 23, 2011)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[579, 22, 617, 17, 30, "Input"],
Cell[1199, 41, 331, 10, 47, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[1567, 56, 241, 6, 30, "Input"],
Cell[1811, 64, 189, 5, 47, "Output"]
}, Open  ]],
Cell[2015, 72, 354, 9, 30, "Input"],
Cell[CellGroupData[{
Cell[2394, 85, 749, 20, 50, "Input"],
Cell[3146, 107, 242, 6, 34, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[3425, 118, 701, 20, 50, "Input"],
Cell[4129, 140, 354, 10, 47, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[4520, 155, 732, 21, 69, "Input"],
Cell[5255, 178, 631, 21, 55, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[5923, 204, 380, 10, 30, "Input"],
Cell[6306, 216, 617, 20, 55, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[6960, 241, 307, 8, 30, "Input"],
Cell[7270, 251, 1100, 35, 58, "Output"]
}, Open  ]],
Cell[8385, 289, 92, 1, 30, "Input"]
}
]
*)

(* End of internal cache information *)
